Macroscopic polarization as a geometric quantum phase: Many-body formulation

During past decades, concepts about the electrostatics of infinite systems have been a challenge for theoretical physicists. In particular, the question of whether the absolute macroscopic polarization or the difference between the polarizations of two states of an insulating crystal is a well-defined bulk property has remained a controversial one. Recently, King-Smith and Vanderbilt, and Resta have provided an approach in terms of the geometric Berrys phase of electronic orbitals in an independent-particle approximation. Here we extend the derivation of Niu and Thouless for quantized charge transport to the case where the quantum adiabatic evolution is noncyclic, and we show how this polarization difference can be written in terms of a Berrys phase for a system with many-body interactions. We also discuss the origin and magnitude of the ``quantum uncertainty'' that appears when a path-independent gauge is used to compute those geometric quantum phases. This geometric viewpoint not only helps us understand the issues raised above but provides a mathematical method to compute polarizations in a many-body framework.